Just think of the following idea:

What's about printing your up-to-date account information of the frequent flyer programme anywhere from the Internet.

Some airlines are showing clearly and in coulour the status you have. If you are member of BA Executive Club your account infor states the coulour of the card, a picture of it and the latest "transactions". ^^^^

If it's the same with your airline, just do it and print it in coulour. This MAY help even presented with a smile and is just more evidence than only an E-Mail. :-:

It's no plastic card or evidence but the lounge dragon would see that you have tried to help her with her work.

Would be fine if you post your experience if it works. Have a nice flight !